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的 is not only used as a possessive suffix (as in 我的). There are other uses, don't be confused.
In case of 你说的是什么, 的 serves as selector by an attribute. Literally, someone asks, "what is it?", but out of all possible contexts picks up a context of the recent speech.

An illustration:

你想买什么样的手机？ What kind of cell phone you'd like to buy?
很贵的。 A very expensive one.

There's another usage of 的, with the meaning of "certainly". See this answer for details.

One other way to look at it: 我的 is not only "my," but can also be, "mine."

我吃饱了，你吃我的吧。

I'm full, so please eat mine.

Just like in English, in Chinese you could end the sentence with either "my <noun>" or "mine". When you hear this sentence and notice no noun or noun-phrase follows 的, this is perfectly OK, and you can translate the 我的 as "mine".

But unlike English, Chinese follows a pattern. You can put other things before the 的, too. So not only can you specify 你说的事情 "The issue you're talking about," you can instead say 你说的 "That which you're talking about," which is awkward in English but concise and natural in Chinese.

你说 -> you say
你说的 -> what you say (or said, depending on context)
你做 -> you do
你做的 -> what you do (or did, depending on context)
你说的和你做的不一致。 -> What you said does not match what you did.
他说的确实在理。 -> What he said really makes sense.
情况比我们估计的更糟。 -> The situation is worse than what we estimated.
你说的是什么？ -> What is what you said?

The full sentence should be "你说的东西是什么？" "东西" is omitted here.
In English, "What did you say" is short for "What things did you say"
"说的" here is used to prescribe something you said, but not heard (听到的), thought (想的) or did (做的).